text: In 1926, Yakov Frenkel suggested that in an ionic crystal like AgI, in thermodynamic equilibrium, a small fraction of the cations, α, are displaced from their regular lattice sites into interstitial positions. He related α with the Gibbs energy for the formation of one mol of Frenkel pairs, ΔG, as α = exp(-ΔG/2RT), where T is temperature and R is the gas constant; for a typical value of ΔG = 100 kJ/mol, α ~ 1 at 100 °C and ~6 at 400 °C. This idea naturally explained the presence of an appreciable fraction of mobile ions in otherwise defect-free ionic crystals, and thus the ionic conductivity in them.
  Frenkel’s idea was expanded by Carl Wagner and Walter Schottky in their 1929 theory, which described the equilibrium thermodynamics of point defects in ionic crystals. In particular, Wagner and Schottky related the deviations from stoichiometry in those crystals with the chemical potentials of the crystal components, and explained the phenomenon of mixed electronic and ionic conduction.
  Wagner and Schottky considered four extreme cases of point-defect disorder in a stoichiometric binary ionic crystal of type AB:
  # Pairs of interstitial cations A and lattice vacancies (Frenkel defects)
  #Pairs of interstitial anions B and lattice vacancies (anti-Frenkel defects)
  #Pairs of interstitial cations A and interstitial anions B with no vacancies
  # Pairs of A and B-type lattice vacancies with no interstitials (Schottky disorder).
  Type-3 disorder does not occur in practice, and type 2 is observed only in rare cases when anions are smaller than cations, while both types 1 and 4 are common and show the same exp(-ΔG/2RT) temperature dependence.
  Later in 1933, Wagner suggested that in metal oxides an excess of metal would result in extra electrons, while a deficit of metal would produce electron holes, i.e., that atomic non-stoichiometry would result in a mixed ionic-electronic conduction.